Towards a new vision of mesh adaptation methods

and its impact on the simulation of PDEs

Loïc Gouarin et Marc Massot

HPC@Maths team
CMAP / CNRS - Ecole polytechnique

Context

The present work is the result of a team work involving

• Thomas Bellotti (CNRS, EM2C - Fédé Maths CS)
• Loïc Gouarin (École polytechnique, CMAP)
• Josselin Massot (École polytechnique, CMAP)
• Pierre Matalon (École polytechnique, CMAP)
• Laurent Séries (École polytechnique, CMAP)
• Christian Tenaud (CNRS, EM2C - Fédé Maths CS)

website of the team

https://github.com/hpc-maths/

Mesh adaptation and error control

Burgers equation - sinus problem

\[ \partial_t u + \partial_x \left ( f(u) \right ) = 0, \quad t \geq 0, \quad x \in \mathbb{R}, \qquad f(u) = \dfrac{u^2}{2}, \]

Consider the Cauchy problem with initial conditions:

\[ u^0(x) = \frac{1}{2} (1+\sin(\pi(x-1))) \quad x \in [-1,1] \]

Adaptive Multiresolution

• Minimum level \(\underline{\ell}\) and maximum level \(\bar{\ell}\).
• Cells: \[ C_{\ell, k}:=\prod_{\alpha=1}^d\left[2^{-\ell} k_\alpha, 2^{-\ell}\left(k_\alpha+1\right)\right] \]
• Finest step: \(\Delta x=2^{-\bar{\ell}}\).
• Level-wise step: \(\Delta x_{\ell}:=2^{-\ell}=2^{\Delta \ell} \Delta x\).

Wavelets

Decomposition of the solution on a wavelet basis [Daubechies, ’88], [Mallat, ’89] to measure its local regularity. “Practical” approach by [Harten, ’95], [Cohen et al., ’03].

Projection operator

Prediction operator at order \(2 s +1\)

\[ {\hat f}_{\ell+1,2 k}={f}_{\ell, k}+\sum_{\sigma=1}^s \psi_\sigma\left({f}_{\ell, k+\sigma}-{f}_{\ell, k-\sigma}\right) \]

Details are regularity indicator \[ {\mathrm{d}}_{\ell, {k}}:={f}_{\ell, {k}}-{\hat{f}}_{\ell, {k}} \]

Let \(f \in W^{\nu, \infty}\) (neigh. of \(C_{\ell, k}\) ), then \[ \left|{\mathrm{d}}_{\ell, k}\right| \lesssim 2^{-\ell \min (\nu, 2 s +1)}|f|_{W^{\min (\nu, 2 s +1), \infty}} \]

Fast wavelet transform:

means at the finest level can be recast as means at the coarsest level + details \[ \begin{array}{rlr} {f}_{\overline{\ell}} & \Longleftrightarrow & \left({f}_{\underline{\ell}}, {{d}}_{\underline{\ell} +1}, \ldots, {d}_{\bar{\ell}}\right)\\ \end{array} \]

Mesh coarsening (static)

Local regularity of the solution allows to select areas to coarsen

\[ {{f}}_{\bar{\ell}} \rightarrow \left({f}_{\underline{\ell}}, {\mathbf{d}}_{\underline{\ell}+1}, \ldots, {\mathbf{d}}_{\bar{\ell}}\right) \rightarrow \left({f}_{\underline{\ell}}, {\tilde{\mathbf{d}}}_{\underline{\ell}+1}, \ldots, \tilde{{\mathbf{d}}}_{\bar{\ell}}\right) \rightarrow {\tilde{{f}}}_{\bar{\ell}} \] \[ \tilde{{\mathrm{d}}}_{\ell, k}= \begin{cases}0, & \text { if } \left|{\mathbf{d}}_{\ell, k}\right| \leq \epsilon_{\ell}=2^{-d \Delta \ell} \epsilon, \quad \rightarrow \quad\left\|{\mathbf{f}}_{\bar{\ell}}-\tilde{{\mathbf{f}}}_{\bar{\ell}}\right\|_{\ell^p} \lesssim \epsilon \\ {\mathrm{d}}_{\ell, k}, & \text { otherwise} \end{cases} \]

Set a small (below \(\epsilon_{\ell}\)) detail to zero \(\equiv\) erase the cell \(C_{\ell, k}\) from the structure

Examples

Equation

\[ f(x) = 1 - \sqrt{\left| sin \left( \frac{\pi}{2} x \right) \right|} \; \text{for} \; x\in[-1, 1] \]

min level	1
max level	12
ε	10-3
compression rate	96.29%
error	0.00053

Time evolution of PDEs

• Finite volumes with global time step \(\Delta t = \Lambda(\Delta x)\)
• Use dynamic mesh refinement

Mesh updated using “old” information at time \(t\) to accommodate the one at time \(t + \Delta t\)

• Propagation of information : add security cells
• Formation of singularities : (regularity index: \(\nu =0\), \(\mu = \min(\nu,2 s +1)\)) refine if \[ \left|{\mathbf{d}}_{\ell, k}\right| \geq \epsilon_{\ell}\,2^{d+\mu} \]

Finite volumes / conservation / order

Flux evaluation at interfaces between levels

Using the prediction operator allows to evaluate fluxes at the same level

Finite volume method

We use a Godunov flux for the small hat problem

Numerical Analysis and Modified Equations

Linear scalar transport equation

In this work, we are concerned with the numerical solution of the Cauchy problem associated with the linear scalar conservation law

\[ \partial_t u(t, x)+V \partial_x u(t, x)=0, \quad(t, x) \in \mathbb{R}^{+} \times \mathbb{R} \]

where \(V\) is the transport velocity, taken \(V>0\) without loss of generality. we consider 1 d problems. The extension to \(2\mathrm{~d}\) / \(3\mathrm{~d}\) problems is straightforward and usually done by tensorization [Bellotti2022] and yields analogous conclusions.

The discrete volumes are \[ C_{\ell, k}:=\left[2^{-\ell} k, 2^{-\ell}(k+1)\right], \quad k \in \{ 0,2^{\ell}-1 \}, \] for any \(\ell \in \{ \underline{\ell}, \bar{\ell} \}\). The measure of each cell at level \(\ell\) is \(\Delta x_{\ell}:=2^{-\ell}\) and we shall indicate \(\Delta x:=\Delta x_{\bar{\ell}}\). The cell centers are \(x_{\ell, k}:=\) \(2^{-\ell}(k+1 / 2)\). Finally, we shall indicate \(\Delta \ell:=\bar{\ell}-\ell\), hence \(\Delta x_{\ell}=2^{\Delta \ell} \Delta x\).

Finite Volume scheme

Finite Volume scheme at the finest level of resolution \(\bar{\ell}\) for any cell of indices \(\bar{k} \in \{ 0,2^{\bar{\ell}}-1 \}\). Explicit schemes read:

\[ \mathrm{v}_{\bar{\ell}, \bar{k}}^{n+1}=\mathrm{v}_{\bar{\ell}, \bar{k}}^n-\frac{\Delta t}{\Delta x}\left(\Phi\left(\mathrm{v}_{\bar{\ell}, \bar{k}+1 / 2}^n\right)-\Phi\left(\mathrm{v}_{\bar{\ell}, \bar{k}-1 / 2}^n\right)\right) \]

where we utilize the same linear numerical flux for the left and the right flux (conservativity) \[ \Phi\left(\mathbf{v}_{\bar{\ell}, \bar{k}-1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha \mathbf{v}_{\bar{\ell}, \bar{k}+\alpha}, \quad \Phi\left(\mathbf{v}_{\bar{\ell}, \bar{k}+1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha v_{\bar{\ell}, \bar{k}+1+\alpha} \]

Modified equations

[Carpentier et al 97] or Cauchy-Kowalewski procedure [Harten et al 87]

\[ \partial_t u\left(t^n, x_{\bar{\ell}, \bar{k}}\right)+V \partial_x u\left(t^n, x_{\bar{\ell}, \bar{k}}\right)=\sum_{h=2}^{+\infty} \Delta x^{h-1} \sigma_h \partial_x^h u\left(t^n, x_{\bar{\ell}, \bar{k}}\right) \]

• Upwind scheme \[ \partial_t u+V \partial_x u=\frac{\Delta x V}{2}(1-\lambda V) \partial_{x x} u+O\left(\Delta x^2\right) \]
• Lax-Wendroff scheme \[ \partial_t u+V \partial_x u=-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+O\left(\Delta x^3\right) \]
• OSMP-3 scheme \[ \partial_t u+V \partial_x u=\frac{\Delta x^3 V}{24}\left(-\lambda^3 V^2+2 \lambda^2 V^2+\lambda V-2\right) \partial_x^4 u+O\left(\Delta x^4\right) \]

How to include MRA I

We introduce the reconstruction operator \(\hat{s}\) instead of \(s\) on the cells \(\left(\bar{\ell}, 2^{\Delta \ell} k+\delta\right)\) for any \(\delta \in \mathbb{Z}\) at the finest level

• \(\hat{s}=s\) : exact local flux reconstruction [Cohen et al. 2003].
• \(\hat{s}=0\) but \(s>0\), direct evaluation or naive evaluation [Hovhannisyan et al 2010].

\[ \mathbf{w}_{\bar{\ell}, \bar{k}}^{n+1}=\mathbf{w}_{\bar{\ell}, \bar{k}}^n-\frac{\Delta t}{\Delta x}\left(\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}+1 / 2}^n\right)-\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}-1 / 2}^n\right)\right) \] \[ \Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}-1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha \hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}+\alpha} \]

How to include MRA II

Let now \((\ell, k) \in S\left(\tilde{\Lambda}^{n+1}\right)\), taking the projection yields the multiresolution scheme

\[ \mathbf{w}_{\ell, k}^{n+1}=\mathbf{w}_{\ell, k}^n-\frac{\Delta t}{\Delta x_{\ell}}\left(\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell}(k+1)+1 / 2}^n\right)-\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell} k-1 / 2}^n\right)\right) \]

\[ \Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell} k-1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha \hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell} k+\alpha} \]

Some information is loss because of the averaging procedure: two different schemes we can consider for the computation of the modified equations.

Theorem

The local truncation error of the reference Finite Volume scheme and the one of the adaptive Finite Volume scheme are the same up to order \(2\hat{s}+1\) included.

Modified equations including MRA

This result establishes at which order the modified equations of the reference scheme are perturbed by the introduction of the adaptive scheme. However, it does not characterize the terms in the modified equations above order \(2\hat{s}+1\) in \(\Delta x\) (symbolic computations).

• Upwind scheme (original) \[ {\color{blue}{% \partial_t u+V \partial_x u=\frac{\Delta x V}{2}(1-\lambda V) \partial_{x x} u+O\left(\Delta x^2\right) }} \]
• Upwind scheme with MRA \[ \begin{array}{lr} \partial_t u+V \partial_x u=\frac{\Delta x V}{2}\left(2^{\Delta \ell}-\lambda V\right) \partial_{x x} u+O\left(\Delta x^2\right), & \text { for } \hat{s}=0 \\ \partial_t u+V \partial_x u=\frac{\Delta x V}{2}(1-\lambda V) \partial_{x x} u-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+& \\ \ \ \ \ \ \ \ \ \ \ \frac{\Delta x^3 V}{24}\left(-3 \Delta \ell 2^{2 \Delta \ell}+2^{2 \Delta \ell}-\lambda^3 V^3\right) \partial_x^4 u+O\left(\Delta x^4\right), & \text { for } \hat{s}=1 \end{array} \]

Modified equations including MRA

This result establishes at which order the modified equations of the reference scheme are perturbed by the introduction of the adaptive scheme. However, it does not characterize the terms in the modified equations above order \(2\hat{s}+1\) in \(\Delta x\) (symbolic computations).

• Lax-Wendroff scheme (original) \[ {\color{blue}{\partial_t u+V \partial_x u=-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+O\left(\Delta x^3\right)}} \]
• Lax-Wendroff scheme with MRA \[ \begin{array}{lr} \partial_t u+V \partial_x u=\frac{\Delta x \lambda V^2}{2}\left(2^{\Delta \ell}-1\right) \partial_{x x} u+O\left(\Delta x^2\right), & \text { for } \hat{s}=0 \\ \partial_t u+V \partial_x u=-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+&\\ \ \ \ \ \ \ \ \ \ \ \frac{\Delta x^3 \lambda V^2}{24}\left(-3 \Delta \ell 2^{2 \Delta \ell}+2^{2 \Delta \ell}-\lambda^2 V^2\right) \partial_x^4 u+O\left(\Delta x^4\right), & \text { for } \hat{s}=1 \end{array} \]

Theoretical results on the global error

Theorem 2

Assume that

• The reference scheme satisfies the restricted stability condition \(\|E\| \leq 1\)
• The Harten-like scheme satisfies the restricted stability condition \(\left\|\bar{E}_{\Lambda}\right\| \leq 1\) for any \(\Lambda\).

Then, for smooth solution, in the limit \(\Delta x \rightarrow 0\) (i.e. \(\bar{\ell} \rightarrow+\infty\) ) and for \(\Delta \underline{\ell}=\bar{\ell}-\underline{\ell}\) kept fixed, we have the error estimate

\[ \left\|\mathbf{v}_{\bar{\ell}}^n-\mathbf{w}_{\bar{\ell}}^n\right\| \leq C_{t r} t^n \Delta x^{2 \hat{s}+1}+C_{m r} \frac{t^n}{\lambda \Delta x} \epsilon \]

where \(C_{t r}=C_{t r}\left(\bar{\ell}-\underline{\ell},\left(\phi_\alpha\right)_\alpha, \lambda, \hat{s}, V\right)\) and \(C_{m r}=C_{m r}\left(\bar{\ell}-\underline{\ell},\left(\phi_\alpha\right)_\alpha, \lambda, \hat{s}, s, V\right)\). \[ \left\|\mathbf{u}_{\bar{\ell}}^n-\mathbf{w}_{\bar{\ell}}^n\right\| \leq C_{r e f} t^n \Delta x^\theta+C_{t r} t^n \Delta x^{2 \hat{s}+1}+C_{m r} \frac{t^n}{\lambda \Delta x} \epsilon \]

Comments on theorem

• The error estimate contains three contributions: the discretization error of the reference scheme, the perturbation error between the reference and the adaptive scheme, and the thresholding error coming from the multiresolution
• The constant \(C_{\mathrm{tr}}\) generally grows exponentially with \(\bar{\ell}-\underline{\ell}\), sometimes also involving linear terms, i.e. \(\hat{s}=1\). We have the following cases:
  • \(\theta<2 \hat{s}+1\). The error of the reference scheme dominates the perturbation introduced by the adaptive scheme \(\left\|\mathbf{u}_{\bar{\ell}}^N-\mathbf{w}_{\bar{\ell}}^N\right\| \leq C_{\mathrm{ref}} T \Delta x^\theta+C_{\mathrm{mr}} \frac{T}{\lambda \Delta x} \epsilon\). A thresholding error of the same order as the reference error \(\epsilon \sim \Delta x^{\theta+1}\).
  • \(\theta=2 \hat{s}+1\). The error of the reference scheme and the perturbation order are comparable (first example!) We have \(\left\|\mathbf{u}_{\bar{\ell}}^N-\mathbf{w}_{\bar{\ell}}^N\right\| \leq\left(C_{\mathrm{ref}}+C_{\mathrm{tr}}\right) T \Delta x^\theta+C_{\mathrm{mr}} \frac{T}{\lambda \Delta x} \epsilon\).
  • \(\theta>2 \hat{s}+1\). The perturbation introduced by the adaptive scheme dominates the error of the reference scheme. Therefore, multiresolution introduces a large perturbation that yields a different convergence rate. We have \(\left\|\mathbf{u}_{\bar{\ell}}^N-\mathbf{w}_{\bar{\ell}}^N\right\| \leq C_{\mathrm{tr}} T \Delta x^{2 \hat{s}+1}+C_{\mathrm{mr}} \frac{T}{\lambda \Delta x} \epsilon\), thus \(\epsilon \sim \Delta x^{2 \hat{s}+2}\) (AMR !)

Assume that for the choice of \(\bar{\ell}-\underline{\ell}\) at hand, we have \(C_{\mathrm{tr}} \sim C_{\mathrm{ref}}\), then w

How to compute fluxes at the finest level

As we have seen, calculating the flux at the finest level is crucial to obtaining good error estimates. But how does this work in practice?

Let’s take a simple example.

How to compute fluxes at the finest level

Let’s imagine that our finest level is 4 levels away from where we are.

How to compute fluxes at the finest level

Now let’s imagine that we want to calculate the flow using this small cell on the left. We use an operator with s = 1, so we need 3 cells on the level below to calculate it. These 3 cells need 4 cells on the level below to be calculated. We can repeat the process indefinitely and we’ll still need 4 cells on the bottom level.

We can determine the coefficients of our linear application to go from our level to any cell on a higher level. There’s no need for an intermediate level.

How to compute fluxes at the finest level

We can do the same for the right-hand cell. We can see that the cells used are not exactly the same.

Now, if we wanted to reconstruct all the cells of the finest level, we’d only need these 5 cells, with different coefficients depending on the cell we’re trying to reconstruct.

Burgers results

Burgers results (Error for scheme order 1)

Burgers results (Error for scheme order 1)

Burgers results (Error for scheme order 1)

Burgers results (MR solution and level for scheme order 1)

Burgers results (MR solution and error for scheme order 1)

Burgers results (MR+MLF sol. and level for scheme order 1)

Burgers results (MR+MLF sol. and error for scheme order 1)

Burgers results (Error for scheme order 1)

Burgers results (MR solution and level for scheme order 1)

Burgers results (MR solution and error for scheme order 1)

Burgers results (MR+MLF sol. and level for scheme order 1)

Burgers results (MR+MLF sol. and error for scheme order 1)

Burgers results (Error for scheme order 3)

Burgers results (Error for scheme order 3)

Burgers results (MR solution and level for scheme order 3)

Burgers results (MR solution and error for scheme order 3)

Burgers results (MR+MLF sol. and level for scheme order 3)

Burgers results (MR+MLF sol. and error for scheme order 3)

Burgers results (MR VS MR+MLF error for scheme order 3)

Burgers 2D results (MR+MLF solution order 1)

Euler 2D results (MR+MLF solution order 3 (OSMP scheme))

Euler 2D results (MR+MLF solution order 3 (OSMP scheme))

Conclusion

• Analysis of mesh adaptation completed for convection - extension to reaction and diffusion
  • Link with AMR important in terms of accuracy of the simulations
• Conducting simulation with strict error control requires adpative multiresolution
  • Data structure is a key issue - see presentation of L. Gouarin
  • New generation of coupled splitting / ImEx schemes in samurai
• Link through the code and developper team to various applications

samurai

Patched-based representation

Advantages

• compact data structure
• rectangular zones work well for tiling and optimizing caches
• efficient to work with neighbors and the different levels

Disadvantages

• requires more cells than necessary
  • grid hierarchy: overlapping
  • larger refinement zone

Cell-based representation

Advantages

• no grids hierarchy
• keep only needed cells

Disadvantages

• more difficult to find the neighbors
• more difficult to work with the different levels
• tree-like structure
• more difficult to add ghost cells hierarchy

If we look at all the open source software specializing in dynamic mesh adaptation, there are two main families: - patch-based, which is a hierarchical representation of the mesh: layers are placed on top of layers - cell-based, which is a flat representation of the mesh

Each has its advantages and disadvantages: - patch-based has rectangular zones for tiling and optimizing caches. But it generally requires more cells than necessary. - cell-based requires far fewer cells but requires a tree-like structure, which means that you lose the good memory locality you had with patch-based. We use space filling curves such as Morton or Hilbert to find an acceptable locality.

Adaptive methods

AMR

Advantages

• based on heuristic criteria (gradient, second derivative, …)
• easy to implement

Disadvantages

• physical problem understanding is important
• add potentially more cells that needed

MRA (see the presentation of M. Massot)

Advantages

• based on wavelet decomposition
• error control
• independent of the physical problem
• only needed cells

Disadvantages

• difficult to implement with the current data structures
• can be costly

Can we have a data structure that takes the advantages of both and well suited to implement various adaptive mesh refinement methods ?

Samurai

Design principles

• Compress the mesh according to the level-wise spatial connectivity along each Cartesian axis.
• Achieve fast look-up for a cell into the structure, especially for parents and neighbors.
• Maximize the memory contiguity of the stored data to enable caching and vectorization.
• Manage and facilitate complex operations between meshes (numerical schemes, MRA, …).

An overview of the data structure

[ start , end [ @ offset

We illustrate the samurai data structure, starting with a 2D problem to see all the components. The data structure is fully recursive and can therefore handle Nd Cartesian meshes.

The main element of samurai is an interval modeled by its beginning and non-included end. There’s also an offset parameter that will serve us in two different ways. We’ll come back to this later.

An overview of the data structure

• Level 0
• x: [0, 4[, [0, 1[, [3, 4[, [0, 1[, [3, 4[, [0, 3[
• y: [0, 4[
• y-offset: [0, 1, 3, 5, 6]
• Level 1
• x: [2, 6[, [2, 6[, [2, 4[, [5, 6[, [2, 6[, [6, 8[, [6, 7[
• y: [2, 8[
• y-offset = [0, 1, 2, 4, 5, 6, 7]
• Level 2
• x: [8, 10[, [8, 10[, [14, 16[, [14, 16[
• y: [8, 10[, [14, 16[
• y-offset: [0, 1, 2, 3, 4]

The first thing we’re going to do is go through each level and each y-component, and look at the set of contiguous cells along the x-axis to construct intervals.

Identify the different types of cells

Identify the different types of cells

Identify the different types of cells

Identify the different types of cells

Algebra of sets

\

=

Algebra of sets

The search of an admissible set is recursive. The algorithm starts from the last dimension (y in 2d, z in 3d,…).

The available operators in samurai are for now

• the intersection of sets,
• the union of sets,
• the difference between two sets,
• the translation of a set,
• the projection of a set.

The search for an admissible set is completely recursive. We’ll start by searching in the largest dimension and if we don’t find anything we’ll stop.

If an admissible interval is found, we look at it in the lower dimension, which means we manipulate very few intervals because we’re only looking at a subset.

Usage example: find jumps

auto jump_set = intersection(
                    translate(mesh[1], {1}).on(0),
                    mesh[0]
                );

Usage example: the projection operator

auto proj_set = intersection(mesh[level], mesh[level + 1])
          .on(level);

proj_set([&](auto i)
{
    u(level, i) = 0.5*(u(level+1, 2*i) + u(level+1, 2*i+1));
});

Compression rates

Compression rates

Level	Num. of cells	p4est	samurai (leaves)	samurai (all)	ratio
\(9\)	66379	2.57 Mb	33.68 Kb	121 Kb	21.24
\(10\)	263767	10.25 Mb	66.64 Kb	236.8 Kb	43.28
\(11\)	1051747	40.96 Mb	132.36 Kb	467.24 Kb	87.66
\(12\)	4200559	163.75 Mb	263.6 Kb	927 Kb	176.64
\(13\)	16789627	654.86 Mb	525.9 Kb	1.85 Mb	353.98
\(14\)	67133575	2.61 Gb	1.05 Mb	3.68 Mb	709.24

We can see here that the mesh is very compressed compared to a version using a tree structure.

The total size of the mesh is multiplied by just under 4, as we build different meshes to handle ghost cells and prediction cells.

What’s important to note is that in a parallel context, it doesn’t cost much for each process to have the meshes of its neighboring domains.

Other features

• Loop algorithms over the levels and the cells
• Simplified access operator
• Reconstruct a value anywhere even if the cell doesn’t exist
• Helper classes to construct complex meshes
• Helper classes to construct schemes for explicit and implicit usage
• Helper classes to construct N-D operators and expressions using xtensor or Eigen
• MPI support
• HDF5 support

Roadmap

Here’s the samurai roadmap for the coming months and probably years.

We now have the possibility of changing the containers in which our fields are stored. We did this so that we could start using Kokkos. So we’d be very interested in working with the people at Cexa on this.

We have received several funding packages that have enabled us to hire two research engineers.

As part of numpex, we also have an engineer position to work on the I/O part of AMR methods in collaboration with Dyablo developers.

People involved

• 4 core developers
• users and developers communities
• Various collaborations and research programs

Real-World Applications

• Interfacial flow simulation
• Simulation analysis on the Hydrogen risk
• Plasma dynamics
• Battery simulation
• Combustion modeling

Samurai

https://github.com/hpc-maths/samurai